Generally only more sophisticated three- to five-star hotels offer a babysitting service. Here is a useful list:
Your best bet is to go for one of the many internet resources that specialise in connecting families with suitable babysitters, such as:
The following webpage provides an extensive list of English preschools, kindergartens, day-care centres and nursery schools in Athens:
As anywhere, networking is important: internet groups and organsiations such as those mentioned above will be useful, but try and get to know parents and working mothers in your area, in the park etc.